Skip to main content
Log in

On Error Bounds for Quasinormal Programs

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The Mangasarian-Fromovitz constraint qualification is a central concept within the theory of constraint qualifications in nonlinear optimization. Nevertheless there are problems where this condition does not hold though other constraint qualifications can be fulfilled. One of such constraint qualifications is the so-called quasinormality by Hestenes. The well known error bound property (R-regularity) can also play the role of a general constraint qualification providing the existence of Lagrange multipliers. In this note we investigate the relation between some constraint qualifications and prove that quasinormality implies the error bound property, while the reciprocal is not true.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Mangasarian, O.L., Fromovitz, S.: The Fritz-John necessary optimality conditions in presence of equality and inequality constraints. J. Math. Anal. Appl. 17, 37–47 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  2. Janin, R.: Directional derivative of the marginal function in nonlinear programming. Math. Program. Study 21, 110–126 (1984)

    MATH  MathSciNet  Google Scholar 

  3. Qi, L., Wei, Z.: On the constant positive linear independence condition and its application to SQP methods. SIAM J. Optim. 10, 963–981 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Andreani, R., Martinez, J.M., Schverdt, M.L.: On the relations between constant positive linear dependence condition and quasinormality constraint qualification. J. Optim. Theory Appl. 125, 473–485 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hestenes, M.R.: Optimization Theory—the Finite Dimensional Case. Wiley, New York (1975)

    MATH  Google Scholar 

  6. Bertsekas, D.P., Ozdaglar, A.E.: The relation between pseudonormality and quasiregularity in constrained optimization. Optim. Methods Softw. 19, 493–506 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Robinson, S.M.: Stability theory for systems of inequalities. Part 2. Differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fedorov, V.V.: Numerical Methods of Maxmin. Nauka, Moscow (1979)

    Google Scholar 

  9. Ioffe, A.D.: Regular points of Lipschitz functions. Trans. Am. Math. Soc. 251, 61–69 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  10. Luderer, B., Minchenko, L., Satsura, T.: Multivalued Analysis and Nonlinear Programming Problems with Perturbations. Kluwer Academic, Dordrecht (2002)

    MATH  Google Scholar 

  11. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  12. Minchenko, L., Stakhovski, S.: On relaxed constant rank regularity condition in mathematical programming. Optimization 60(3) (2011). doi:10.1080/02331930902971377

  13. Abadie, J.M.: On the Kuhn-Tucker theorem. In: Abadie, J. (ed.) Nonlinear Programming, pp. 19–36. North-Holland, Amsterdam (1967)

    Google Scholar 

  14. Giannessi, F.: Constrained Optimization and Image Space Analysis, Separation of Sets and Optimality Conditions, vol. 1. Springer, New York (2005)

    Google Scholar 

  15. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 1: sufficient optimality conditions. J. Optim. Theory Appl. 142, 147–163 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Moldovan, A., Pellegrini, L.: On regularity for constrained extremum problems. Part 2: necessary optimality conditions. J. Optim. Theory Appl. 142, 165–183 (2009)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to L. Minchenko.

Additional information

Communicated by F. Giannessi.

This research was supported by Belorussian Republican Foundation for Fundamental Research and State Programs for Fundamental Research “Mathematical Models” and “Mathematical Methods”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Minchenko, L., Tarakanov, A. On Error Bounds for Quasinormal Programs. J Optim Theory Appl 148, 571–579 (2011). https://doi.org/10.1007/s10957-010-9768-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-010-9768-0

Keywords

Navigation